There are many inventions described and illustrated herein. The inventions relate to microelectromechanical and/or nanoelectromechanical (collectively hereinafter “microelectromechanical”) structures and devices/systems including same; and more particularly, in one aspect, to oscillator systems employing microelectromechanical resonating structures, and methods to control and/or operate same.
Microelectromechanical systems, for example, gyroscopes, oscillators, resonators and accelerometers, utilize micromachining techniques (i.e., lithographic and other precision fabrication techniques) to reduce mechanical components to a scale that is generally comparable to microelectronics. Microelectromechanical systems typically include a microelectromechanical structure fabricated from or on, for example, a silicon substrate using micromachining techniques. The operation and the response of the microelectromechanical structures depend, to a significant extent, on the operating temperature of the structure.
Where the microelectromechanical system is, for example, a resonator, which is fabricated from or on silicon, the performance of the microelectromechanical resonator is dependent on the operating temperature of the resonator. In this regard, temperature fluctuations may result in, for example, changes in (i) microelectromechanical resonator geometry, (ii) microelectromechanical resonator mass, (iii) stresses or strains on the microelectromechanical resonator (for example, changes in stresses/strains due to, among other things, the thermal coefficient of expansion of the resonator, substrate and/or packaging (if any)), and (iv) the material properties of the resonator. Among thermally-induced changes, the elastic sensitivity of silicon to temperature often dominates in many silicon-based microelectromechanical resonator designs, which often results in a resonator frequency shift in the range of about −20 ppm/C to about −30 ppm/C.
As is well understood, the Young's modulus for most materials of interest changes with temperature according to known thermal coefficients. For example, polysilicon has a first-order thermal coefficient of −75 ppm/C. Furthermore, the geometry of a beam structure also changes with temperature, generally expanding with increasing in temperature. Again, as an example, polysilicon has a thermal expansion coefficient of 2.5 ppm/C.
For some beam designs and related modeling purposes, and given a material with an isotropic thermal coefficient of expansion, the effect of thermal expansion of the width of the beam is somewhat offset by the effect of thermal expansion of the length of the beam. While it may be possible to compensate for some thermally-induced changes in the resonator based on the coefficient of thermal expansion, the shift in Young's modulus over temperature generally dominates in many resonator designs.
Setting aside electrostatic forces, the resonance frequency (f) of a beam may be characterized under these assumptions by the equation:
  f  =            1              2        ⁢                                  ⁢        π              ⁢                            k          eff                          m          eff                    where keff is the effective stiffness of the beam, and meff is the effective mass of the beam which is often constant over temperature.
In most implementations, the resonance frequency of the microelectromechanical resonator should remain substantially stable over a range of operating temperatures. This, however, will not normally be the case as thermally induced changes to the Young's modulus (or other variables) tend to change in the mechanical stiffness of the beam. As such, thermally-induced changes to the Young's modulus tend to cause considerable drift or change in the frequency of the output of the resonator. (See, for example, FIG. 1).
Given typical requirements for temperature stabilities ranging in magnitude from 0.1 to 100 ppm, and common operating temperature specifications ranging from −40 C to +85 C, there is a need for a temperature stable frequency (over an operating temperature range) of output signals of systems employing microelectromechanical resonators.